

A117111


Sum of four positive heptagonal numbers A000566.


2



4, 10, 16, 21, 22, 27, 28, 33, 37, 38, 39, 43, 44, 49, 50, 54, 55, 58, 60, 61, 64, 66, 70, 71, 72, 75, 76, 77, 81, 82, 84, 87, 88, 90, 91, 92, 93, 96, 97, 98, 101, 102, 103, 104, 107, 108, 109, 112, 113, 114, 115, 117, 118, 120, 121, 123, 124, 125, 127, 129, 130, 132
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OFFSET

1,1


COMMENTS

Fermat discovered, Gauss, Legendre and [1813] Cauchy proved that every integer is the sum of 7 heptagonal numbers (and there are some numbers which require all 7, the smallest being 13). 7 is the only prime heptagonal number. Primes which are sums of two positive heptagonal numbers include: {19, 41, 73, 89, 113, 149, 167, 193, 223, 229, 269, 293, 337, 347, 367, 383, 521, ...}. Primes which are sums of three positive heptagonal numbers include: {3, 37, 43, 53, 59, 83, 89, 107, 131, 137, 149, 163, 167, 173, 191, 197, 211, 227, 241, 251, 257, 263, 271, ...}. Primes which are sums of four positive heptagonal numbers include: {37, 43, 61, 71, 97, 101, 103, 107, 109, 113, 127, 149, 151, 167, 181, 191, 197, 199, 211, 223, 229, 239, 251, ...}.


LINKS

Harvey P. Dale, Table of n, a(n) for n = 1..2000


FORMULA

{a(n)} = {A000566} + {A000566} + {A000566} + {A000566} = {a*(5*a3)/2 + b*(5*b3)/2 + c*(5*c3)/2 + d*(5*d3)/2 such that every term is positive}.


MATHEMATICA

Module[{upto=150, max}, max=Ceiling[(3+Sqrt[9+40upto])/10]; Select[Total/@
Tuples[PolygonalNumber[7, Range[max]], 4]//Union, #<=upto&]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 15 2016 *)


CROSSREFS

Cf. A000566, A000040, A000326, A003679, A064826, A117065.
Sequence in context: A223961 A224391 A224024 * A310508 A310509 A310510
Adjacent sequences: A117108 A117109 A117110 * A117112 A117113 A117114


KEYWORD

easy,nonn


AUTHOR

Jonathan Vos Post, Apr 18 2006


STATUS

approved



